On Finite Essential Extensions of Torsion Free Abelian Groups

Abstract

Let A be a torsion free abelian group. We say that a group K is a finite essential extension of A if K contains an essential subgroup of finite index which is isomorphic to A. Such K admits a representation as (A ℤ xkx)/ℤky where y = Nx + a for some k x k matrix N over Z and α ∈ Ak satisfying certain conditions of relative primeness and ℤk = {(α1,..., αk) : αi, ∈ ℤ}. The concept of absolute width of an f.e.e. K of A is defined and it is shown to be strictly smaller than the rank of A. A kind of basis substitution with respect to Smith diagonal matrices is shown to hold for homogeneous completely decomposable groups. This result together with general properties of our representations are used to provide a self contained proof that acd groups with two critical types are direct sum of groups of rank one and two.

Representations over PID’s with three distinguished submodules

“…The following consequence of Corollary 4.6 has been noted in a number of recent papers [5,9,28,22,46]. It provides, in effect, a complete classification of finite-rank Butler groups with at most two critical types.…”

Gauß' theorem for two submodules

“…The finite rank members of B(T 2 , ω) have received attention in a number of recent papers [3,6,10,64,67]. Each account provides its own proof of the following result, which has been dubbed the "small typeset theorem.…”

Stacked Bases for Countable Homogeneous Completely Decomposable Groups

“…In fact they proved a stronger result saying that in essence every canonical decomposition of X=A into torsion cyclic groups can be lifted to a simultaneous decomposition of A and X (see [9] and [10]). In a recent paper K. Benabdallah and the ®rst author proved, using an approach from linear algebra, that a pair A X of hcd groups of ®nite rank has stacked bases if the quotient X=A is a direct sum of torsion cyclic groups.…”